In this paper, we propose a new family of methods for graph learning with low-rank constraints on the covariance matrix, hereafter referred to as graphical factor models (GFM).
First, we reformulate graph learning as a problem that encompasses both elliptical distributions and low-rank factor models.
The main interest of generalizing Gaussian graphical models to elliptical ones is to ensure robustness to underlying heavy-tailed distributions [1]}, [2]}, [3]}, [4]}. Moreover, additionally considering low-rank factor models allows for an effective dimensionality reduction.
The main novelty of our approach is to tackle the resulting class of constrained and penalized maximum likelihood estimation in a unified way with Riemannian optimization [5]}, [6]}.
To do so, we leverage geometries of of both the positive definite matrices [7]}, and positive semi-definite matrices of fixed rank [8]}, [9]} that are well suited to the considered models.
The corresponding tools allows us to develop optimization methods that ensure the desired structures for the covariance matrix, thus providing a flexible framework for learning graphical factor models.
